With this prescription, the KdV evolution now induces a nontrivial dynamics \(\mathfrak {U}_t x_i^0 = x_i^t\) . This dynamics has an explicit algorithmic implementation as the “flea gas" [1]}, but its microscopic realisation is not important here. It satisfies the asymptotic condition \(x_i^t \sim x_i^- + 4\eta _i^2 t\)  (\(t\rightarrow -\infty \) ). Using this, we may relate the coordinates \(x_i^0\)  of the quasi-particles at time 0, to the asymptotic coordinates \(x_i^-\) , taking into account all 2-soliton scattering events that occurred at negative times. As such, the quasi-particle gas, with coordinates \(x_i^0\) , is akin to the Bethe wave function (REF ).
